Theorem rspesbca 3039

Description: Existence form of rspsbca 3038. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspesbca	⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbca

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2958	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	rspcev 2834	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
3		cbvrexsv 2713	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
4	2, 3	sylibr 133	1 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ∧ wa 103  [wsb 1755   ∈ wcel 2141  ∃wrex 2449  [wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956

This theorem is referenced by:  spesbc  3040